Free $(\ell r, rr)$-dibands

We prove that varieties of $(\ell r, rr)$-dibands and $(\ell n, rn)$-dibands coincide and describe the structure of free $(\ell r, rr)$-dibands. We also show that operations of an idempotent dimonoid with left (right) regular bands coincide, construct a new class of dimonoids and for such dimonoids give an example of a semiretraction.